So I just finished my fourth year studying for an integrated Masters degree at Cambridge. I really struggled with mental health during the fourth year and ended up convinced that my project (which was 50% of my grade) was a total failure and so I ended up not doing the viva or any of my exams on the taught content. My project was subsequently marked and given a 2.1 so I would have been fine but I was not in a fit state to do my exams and by this point I had decided to intermit and repeat the year (which would have been a struggle as everyone I know would have graduated). However I had a PhD offer at another university and they are happy for me to start in September without having to repeat the year (Cambridge does not give an overall grade for the integrated masters degree so I will graduate with a 2.1 in bachelor's and 'deemed to deserve honours' in my masters - it is a grey area what grade I can declare for my degree).
However I am now having massive doubts about my capability to carry out the PhD - it is 300 miles away from most of my family and friends and although I initially found the topic interesting, because of all that has happened I no longer have any faith in my capacity to do research, and do not really want to work in academia long term. Unfortunately I have now lost my chance to repeat the year and my job prospects look bleak due to what is essentially a failed masters year attached to my bachelor's - in addition there aren't really any jobs I am interested in. Is it worth attempting to go and do the PhD anyway and seeing what happens? I am worried about moving to a new city where I know very few people and I know PhDs are very stressful. However it was challenging to get the offer in the first place and I would like to make the most of it!

Things are doing not that bad, you have a degree from Cambridge! Your project was 2:1 despite you not doing a viva which I would consider a success and you even got offered a PhD. You are definitely capable even if you don't believe it. Though have you talked with people in real life about your stress and mental health problems? Mental health problems can keep getting worse unless you address the problem and they can destroy your self confidence in the process. I would focus on sorting your mental health before making any major decisions.

Don't do a PhD just for the sake of doing one. If you have no interest in academia or the project, what is the point. You are just setting yourself up to fail.

Agree with rewt here.
A PhD emotionally and mentally taxes the strongest amongst us.
I would never recommend undertaking one unless you sort out your mental health issues first.
In particular, I would be very concerned that your response, to what you thought was a poor Masters project, was to avoid the viva and your other exams.
This tells me you are not in a good place at all and I'm afraid it is highly likely that if your PhD hits problems (which it will) that you will struggle to cope.
You need a good base camp from which to explore your research topic. After you have that, sure go ahead.